Xianfeng Gu

Geometry images

Surface geometry is often modeled with irregular triangle meshes. The process of remeshing refers to approximating such geometry using a mesh with (semi)-regular connectivity, which has advantages for many graphics applications. However, current techniques for remeshing arbitrary surfaces create only semi-regular meshes. The original mesh is typically decomposed into a set of disk-like charts, onto which the geometry is parametrized and sampled. In this paper, we propose to remesh an arbitrary surface onto a completely regular structure we call a geometry image. It captures geometry as a simple 2D array of quantized points. Surface signals like normals and colors are stored in similar 2D arrays using the same implicit surface parametrization --- texture coordinates are absent. To create a geometry image, we cut an arbitrary mesh along a network of edge paths, and parametrize the resulting single chart onto a square. Geometry images can be encoded using traditional image compression algorithms, such as wavelet-based coders.

Global conformal surface parameterization

We solve the problem of computing global conformal parameterizations for surfaces with nontrivial topologies. The parameterization is global in the sense that it preserves the conformality everywhere except for a few points, and has no boundary of discontinuity. We analyze the structure of the space of all global conformal parameterizations of a given surface and find all possible solutions by constructing a basis of the underlying linear solution space. This space has a natural structure solely determined by the surface geometry, so our computing result is independent of connectivity, insensitive to resolution, and independent of the algorithms to discover it. Our algorithm is based on the properties of gradient fields of conformal maps, which are closedness, harmonity, conjugacy, duality and symmetry. These properties can be formulated by sparse linear systems, so the method is easy to implement and the entire process is automatic. We also introduce a novel topological modification method to improve the uniformity of the parameterization. Based on the global conformal parameterization of a surface, we can construct a conformal atlas and use it to build conformal geometry images which have very accurate reconstructed normals.

Discrete Surface Ricci Flow

This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary topologies by user-defined Gaussian curvatures. Furthermore, the target metrics are conformal (angle-preserving) to the original metrics. A Ricci flow conformally deforms the Riemannian metric on a surface according to its induced curvature, such that the curvature evolves like a heat diffusion process. Eventually, the curvature becomes the user defined curvature. Discrete Ricci flow algorithms are based on a variational framework. Given a mesh, all possible metrics form a linear space, and all possible curvatures form a convex polytope. The Ricci energy is defined on the metric space, which reaches its minimum at the desired metric. The Ricci flow is the negative gradient flow of the Ricci energy. Furthermore, the Ricci energy can be optimized using Newtons method more efficiently. Discrete Ricci flow algorithms are rigorous and efficient. Our experimental results demonstrate the efficiency, accuracy and flexibility of the algorithms. They have the potential for a wide range of applications in graphics, geometric modeling, and medical imaging. We demonstrate their practical values by global surface parameterizations.

Optimal Global Conformal Surface Parameterization

All orientable metric surfaces are Riemann surfaces and admit global conformal parameterizations. Riemann surface structure is a fundamental structure and governs many natural physical phenomena, such as heat diffusion and electro-magnetic fields on the surface. A good parameterization is crucial for simulation and visualization. This paper provides an explicit method for finding optimal global conformal parameterizations of arbitrary surfaces. It relies on certain holomorphic differential forms and conformal mappings from differential geometry and Riemann surface theories. Algorithms are developed to modify topology, locate zero points, and determine cohomology types of differential forms. The implementation is based on a finite dimensional optimization method. The optimal parameterization is intrinsic to the geometry, preserves angular structure, and can play an important role in various applications including texture mapping, remeshing, morphing and simulation. The method is demonstrated by visualizing the Riemann surface structure of real surfaces represented as triangle meshes.

Conformal virtual colon flattening

We present an efficient colon flattening algorithm using a conformal structure, which is angle-preserving and minimizes the global distortion. Moreover, our algorithm is general as it can handle high genus surfaces. First, the colon wall is segmented and extracted from the CT data set of the abdomen. The topology noise (i.e., minute handle) is located and removed automatically. The holomorphic 1-form, a pair of orthogonal vector fields, is then computed on the 3D colon surface mesh using the conjugate gradient method. The colon surface is cut along a vertical trajectory traced using the holomorphic 1-form. Consequently, the 3D colon surface is conformally mapped to a 2D rectangle. The flattened 2D mesh is then rendered using a direct volume rendering method accelerated with the GPU. Our algorithm is tested with a number of CT data sets of real pathological cases, and gives consistent results. We demonstrate that the shape of the polyps is well preserved on the flattened colon images, which provides an efficient way to enhance the navigation of a virtual colonoscopy system.

Genus zero surface conformal mapping and its application to brain surface mapping.

We developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one-forms for surfaces with or without boundaries (Gu and Yau, 2002), (Gu and Yau, 2003). For genus zero surfaces, our algorithm can find a unique mapping between any two genus zero manifolds by minimizing the harmonic energy of the map. In this paper, we apply the algorithm to the cortical surface matching problem. We use a mesh structure to represent the brain surface. Further constraints are added to ensure that the conformal map is unique. Empirical tests on magnetic resonance imaging (MRI) data show that the mappings preserve angular relationships, are stable in MRIs acquired at different times, and are robust to differences in data triangulation, and resolution. Compared with other brain surface conformal mapping algorithms, our algorithm is more stable and has good extensibility.

Meshless thin-shell simulation based on global conformal parameterization

This paper presents a new approach to the physically-based thin-shell simulation of point-sampled geometry via explicit, global conformal point-surface parameterization and meshless dynamics. The point-based global parameterization is founded upon the rigorous mathematics of Riemann surface theory and Hodge theory. The parameterization is globally conformal everywhere except for a minimum number of zero points. Within our parameterization framework, any well-sampled point surface is functionally equivalent to a manifold, enabling popular and powerful surface-based modeling and physically-based simulation tools to be readily adapted for point geometry processing and animation. In addition, we propose a meshless surface computational paradigm in which the partial differential equations (for dynamic physical simulation) can be applied and solved directly over point samples via moving least squares (MLS) shape functions defined on the global parametric domain without explicit connectivity information. The global conformal parameterization provides a common domain to facilitate accurate meshless simulation and efficient discontinuity modeling for complex branching cracks. Through our experiments on thin-shell elastic deformation and fracture simulation, we demonstrate that our integrative method is very natural, and that it has great potential to further broaden the application scope of point-sampled geometry in graphics and relevant fields

Manifold t-spline

This paper develops the manifold T-splines, which naturally extend the concept and the currently available algorithms/techniques of the popular planar tensor-product NURBS and T-splines to arbitrary manifold domain of any topological type. The key idea is the global conformal parameterization that intuitively induces a tensor-product structure with a finite number of zero points, and hence offering a natural mechanism for generalizing the tensor-product splines throughout the entire manifold. In our shape modeling framework, the manifold T-splines are globally well-defined except at a finite number of extraordinary points, without the need of any tedious trimming and patching work. We present an efficient algorithm to convert triangular meshes to manifold T-splines. Because of the natural, built-in hierarchy of T-splines, we can easily reconstruct a manifold T-spline surface of high-quality with LOD control and hierarchical structure.

Volumetric harmonic brain mapping

We developed two different techniques to study volume mapping problem in Computer Graphics. The first one is to find a harmonic map from a 3 manifold to a 3D solid sphere and the second is a sphere carving algorithm which calculates the simplicial decomposition of volume adapted to surfaces. In this paper, we apply these two techniques to brain mapping problem. We use a tetrahedral mesh to represent the brain volume. The experimental results on both synthetic and brain volume data are reported. We suggest that 3D harmonic mapping of brain volumes to a solid sphere can provide a canonical coordinate system for feature identification and segmentation, as well as anatomical normalization.

Generalized Koebe's method for conformal mapping multiply connected domains

Surface parameterization refers to the process of mapping the surface to canonical planar domains, which plays crucial roles in texture mapping and shape analysis purposes. Most existing techniques focus on simply connected surfaces. It is a challenging problem for multiply connected genus zero surfaces. This work generalizes conventional Koebes method for multiply connected planar domains. According to Koebes uniformization theory, all genus zero multiply connected surfaces can be mapped to a planar disk with multiply circular holes. Furthermore, this kind of mappings are angle preserving and differ by Möbius transformations. We introduce a practical algorithm to explicitly construct such a circular conformal mapping. Our algorithm pipeline is as follows: suppose the input surface has n boundaries, first we choose 2 boundaries, and fill the other n -- 2 boundaries to get a topological annulus; then we apply discrete Yamabe flow method to conformally map the topological annulus to a planar annulus; then we remove the filled patches to get a planar multiply connected domain. We repeat this step for the planar domain iteratively. The two chosen boundaries differ from step to step. The iterative construction leads to the desired conformal mapping, such that all the boundaries are mapped to circles. In theory, this method converges quadratically faster than conventional Koebes method. We give theoretic proof and estimation for the converging rate. In practice, it is much more robust and efficient than conventional non-linear methods based on curvature flow. Experimental results demonstrate the robustness and efficiency of the method.